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Circuit Surgery Regular clinic by Ian Bell Topics in digital signal processing – sampling T his month, we will start to look at various topics related to digital signal processing (DSP). DSP covers a wide range of electronics applications where real-world signals such as sound, images, temperature, pressure, position and speed are converted to digital data for manipulation, analysis, storage and display. The resulting processed digital information, or new information which has been generated digitally, can also be converted into realworld signals for human consumption (eg, audio and video) or manipulation of the physical world (eg, mechanical and thermal). These systems can be relatively complex (for example, streaming movies from the internet or control of industrial robots) or relatively simple (for example, a basic microcontroller measuring temperature). Fig.1 shows the key elements of DSP systems, with a signal path from analogue input via digital processing to analogue output. The input signal passes through an analogue antialiasing filter which removes frequencies that would otherwise cause errors. A sample-andhold circuit captures the filtered input signal at the points in time at which it is to be digitised. The process of sampling, and the use of antialiasing filters are key concepts in DSP and will be discussed in more detail in this article. An analogueto-digital converter (ADC) produces digital codes that represents the values of the analogue signal at the input sampling instants. The digital processing performs operations on the digitised signal (data); this can be implemented directing in digital hardware or by software running on a processor. The result of the digital processing of the input signal (or results of direct digital generation) are converted to analogue values at the output sample times using a digital-to-analogue converter (DAC). The output of the DAC is a ‘stepped’ waveform which is converted to a smooth analogue waveform by the reconstruction filter. Fig.1 shows key components in a generic DSP system, but not all DSP systems will contain all these components. For example, digital generation does not require the input half from Fig.1 and the system’s output may not require a DAC if it is just one or more on/off controls, or a pulse-width-modulated signal. Sampling and quantisation We assume that analogue signals can have an infinite number of possible values between their upper and lower limits (peak values). We also assume that their value may be different (by some arbitrary small amount) after an infinitely small change in time. This is not strictly true when you consider that electric current depends on discrete charge carries (eg, electrons) and quantum effects come in to play when at sufficiently small scales. However, for the majority of analogue electronic circuit designs it is reasonable to assume this property for analogue signals. On the other hand, digitised analogue signals can only have a finite number of possible values. This is because they are represented by digital code values, which have a finite number of bits. An ADC/DAC with an N-bit output/ input digital word can handle at most 2N different values. When an analogue signal is digitised, the process of going from an effectively infinite number of possible values to a finite one is called quantisation and is an inherent property of ADCs. We will look at ADCs (and DACs) in more detail in a later article. Digitised (sampled) analogue signals also only change to a new value after a finite amount of time – this is the fundamental nature of sampling, where the signal value is captured and stored for processing at specific points in time. Sampling usually occurs with a fixed time interval (called the sample period, T) between sample points. A sampling period of T corresponds with a sampling frequency (or rate) of 1/T. Sampling frequency is a key performance parameter of DSP systems. In the most basic approaches, the sampling frequency is the same throughout the system, but it is also possible to use different rates in different parts of a system. For example, initially sampling the analogue signal at a relatively high rate, but effectively resampling (within the digital part) to a lower rate for performing the processing operations (this is called ‘downsampling’ or ‘decimation’). A signal defined at a set of sample times is referred to as a discrete time signal, to distinguish it from a continuous time signal. Digital (DSP) circuits process discrete time signals, but it is also possible to build analogue circuits which work with sampled signals. These use continuousvalue analogue signals that update to new values at each sample point. Switchedcapacitor circuits are an important example of this technology, and are widely used in integrated circuit design. Sampled signals Fig.1. Generic DSP system structure. The top plot in Fig.2 shows a set of sample points, with sample period T, on an analogue waveform. The sampled signal itself, in its most general form, is just a set of values associated with the sample times. It does not have a value between these times – this is illustrated in the middle waveform in Fig.2. This representation fits directly with digital data – a set of discrete values stored in a digital circuit, computer memory or other media. If sampled values are processed in an analogue circuit, then the signal is often held constant between sample points and waveforms typically have a stepped appearance, like the bottom Out plot in Fig.2. This is the waveform that would be seen at the output of both the sample-and-hold and Practical Electronics | May | 2024 63 Analogue In Antialiasing filter Sample and hold Digital ADC Digital processing Analogue DAC Reconstruction filter Continuous waveform with sample points T Samples Sample and hold Fig.2. Sampled waveforms. DAC blocks of the generic DSP system shown in Fig.1. The input waveform is sampled at each sample time and held at this value until the next sample time. At the ADC input the flat parts of the waveform provide the ADC with a constant input during the time it takes to perform the conversion. If the value being converted is not exactly equal to the signal at the sample time then errors will be introduced into the digitised signal. The stepped waveform (particularly at the DAC output) is a called a zeroorder hold. It represents the creation, or reconstruction, of a continuous-in-time analogue signal from a discrete set of samples. This is an interpolation process – we need to fill in the gaps between the sample points to create a waveform that is continuous-in-time. This is similar to finding the ‘line of best fit’ for data points on a graph. The term ‘zero order’ refers to the mathematical principle of approximating a function (or set of data points) with a polynomial. A polynomial function of x is of the form a0+a1x+a2x2+a3x3… and so on, where the an terms are numerical values. The order of the polynomial is the highest power of x used. A zero-order polynomial just has a constant value – a flat line that does not change with x (or t for time in the case of the waveform). It is possible to create higher-order s(t) xC(t) X xs(t) = s(t)xC(t) Fig.3. Simple model of sampling a signal with a pulse train. 64 ‘holds’; for example, p Area = p x 1 a first-order hold 1 produces a straightline value change t 0 between sample –1T 1T 2T 3T t i m e s . H o w e v e r, these are relatively Fig.4. Sampling pulse train s(t). rare/advanced techniques in DSP systems. contains frequencies relatively close to If we sample an analogue the sampling frequency. signal, convert it to digital Moiré aliasing patterns occur when t with an ADC and then convert a digital camera has trouble imaging it directly back to an analogue an intricate pattern, the result is signal with a DAC, the visual artefacts in photos or videos. resulting zero-hold signal is A well-known example is a finely not an accurate reconstruction striped or patterned shirt (eg, on a TV of the original signal, However, presenter) causing strange waves or assuming the sampling swirl patterns to appear in a digital t was performed correctly, image. This happens when the pixel the original signal can be size is similar to the pattern size and reconstructed by passing the is less common than it used to be due DAC output through a lowto the increase in resolution of digial pass filter – the reconstruction cameras and video presentation. filter shown in Fig.1. Any sampling process can be affected by potential ambiguities. This includes manually making real-world Aliasing measurements (eg, of temperature). The first block in Fig.1 is called the Imagine measuring the temperature in ‘antialiasing filter’, so an obvious a garden once a day in the afternoon question is, what is aliasing and why and plotting the results. The shape of do we need a filter for it? Aliasing is a the graph is likely to be an inaccurate problem which can potentially occur representation of the temperature in any signal processing system which changes. For example, there is likely to samples the signal. This includes all be a daily cycle with lower temperatures digital audio and video signal processing. at night. An actual near-constant Digital imaging sensors such as cameras temperature and strong day-night and scanners can also suffer from aliasing cycle with similar temperature each effects. Aliasing is so called because it afternoon will produce the same form of causes ambiguities in the sampled data, graph with this measurement schedule. that is, there could be more than one Measuring every hour would give a signal which resulted in the same set much more accurate picture of what of sample values – thus one signal is an was happening. The choice of sample alias of the other. Aliasing effects can rate requires an understanding of the be observed in everyday life; one oftensignal being sampled. quoted example is the ‘wagon wheel effect’ where the wheels of a vehicle in a film or video (perhaps a stagecoach Sampling concepts in a western, as the name suggests) and nomenclature appear to be turning at the wrong speed, An idealised sampled signal (x s (t)), backwards or even stationary. Search which only exists at the sample points, online for ‘wagon wheel effect’ videos can be approximated by multiplying the if you would like to see some examples. continuous signal (xc(t)) by a train of A video camera samples the scene in short-duration, unit-amplitude pulses front of it at a certain number of frames (s(t)). This is a conceptual model rather per second. If the wheel of a vehicle than practical implementation and is in the shot rotates an exact number of illustrated in Fig.3. The pulse train whole revolutions in the time between signal is shown in Fig.4. The duration frames it will appear to be in the same of each pulse (p) is ideally zero. position in each frame. When the film There is a potential problem here in is viewed the wheel will appear to be that a unit amplitude pulse of duration stationary despite the fact that the vehicle zero simply disappears. As p reduces to is obviously in motion. However, if the zero the area under the pulse on the graph wheel was rotating much more slowly, (p × 1 – see shading on Fig.4) becomes or if the video frame rate was much zero. Fortunately, there is a mathematical higher, we would get a large number solution – a function called the unit of frames per revolution and playback impulse or Dirac delta (written as d(t)) would look as it should. This leads us which is defined to be zero everywhere to the insight that problems will occur except at time zero, but does ‘exist’ in if the input (the scene in this case) the sense that the area under the Dirac Practical Electronics | May | 2024 Fig.5. LTspice schematic to create a sampled waveform. which is what we generally refer to as a waveform. The symbol t is taken to refer to continuous time, and the original continuous signal (xc(t)) has defined values for all values of t. However, for the sampled signal the value is only defined at integer (n) multiples of T, that is nT, so we can write xs(t) as x(nT). With the ‘s’ subscript being less necessary as nT implies sampling. Since T represents the time difference between samples it is also sometime written as DT, the symbol D (upper-case delta) meaning a difference between values of a variable. In its most abstract form, a set of samples is just a sequence of values, and we can refer to these using an index for the position in the sequence, that is x(n) rather than x(nT). For example, we can describe calculations performed on sampled data, where each step occurs at the sampling interval, just using x(n) and other relative samples in the sequence such as x(n – 1) and x(n + 1) to reference the values used in the calculation. If you look at websites and books on DSP you will often see terms such as x(nT), x(nDT), x(n) and d(t) and variants of these in the equations used in these sources. LTspice sampling pulses Fig.6. Results from simulating the LTspice schematic in Fig.5. delta pulse is not zero, despite its zero duration. The area under the Dirac delta is by definition, a value of one. This is a useful mathematical abstraction which occurs frequently in the mathematics of sampled signal processing and is widely used in other areas of physics and engineering. It is worth being aware of the impulse concept even if you do not want to get into all the advanced maths that it gets used in. The sampling pulses occur at multiples of sampling period T; that is, there is a pulse at time t = 0, t = T, t = 2T, t = 3T and so on. There can also be pulses before the nominal reference time zero (t = 0), at times t = –T, t = –2T and so on. In general, mathematical analysis of signals often assumes they are of infinite duration. The term x(t) means the value of the signal x as a function of time (t), We can create sampled signals based on the model illustrated in Fig.3 and Fig.4 using LTspice. This can help us illustrate sampling concepts such as aliasing. The LTspice model in Fig.5 creates a 1kHz sinewave using source V1. A 9kHz sampling pulse train is created using source V2. The sinewave and sampling pulses are multiplied by behavioural source B1 to produce the samples signal in the manner shown in Fig.3. The pulse width of the sampling pulses is chosen to be small but easily visible on the plots because our main aim here is to create illustrative graphs. The result of the simulation is shown in Fig.6. The top trace (red) is the 1kHz sinewave. The second trace (green) shows the sampling pulse train and the third (cyan) the sampled waveform. The bottom trace has the original sinewave and sampled waveform shown together to confirm that the sample pulses align with the original waveform. Fig.7 shows a couple more sinewave signals added to the schematic in Fig.5; Introduction to LTspice Fig.7. Adding more sinewaves to the schematic in Fig.5. Practical Electronics | May | 2024 Want to learn the basics of LTspice? Ian Bell wrote an excellent series of Circuit Surgery articles to get you up and running, see PE October 2018 to January 2019, and July/August 2020. All issues are available in print and PDF from the PE website: https://bit.ly/pe-backissues 65 Fig.8. Results from simulating the LTspice schematic in Fig.7. All three waveforms will produce the same set of samples and are therefore indistinguishable after sampling. Fig.9. Sampling at exactly the Nyquist rate. these are 8kHz and 17kHz. The simulation results in Fig.8 show that the sample pulses from sampling the 1kHz sinewave at 9kHz exactly line up with the other two sinewaves. That means that if we sample a 1kHz, 8kHz or 17kHz sinewave at 9kHz we get the same set of samples. Thus, once we only have the sample data (further through the signal processing system, or in a digital storage), we cannot tell if the data was originally due to a 1kHz, 8kHz or 17kHz sinewave. In fact, there are many more sinewaves that will also produce the same sample data with 9kHz sampling. Another way to look at this is that if we sampled an 8kHz sinewave at 9kHz then when the signal was reconstructed (eg, when playing a digital audio recording) we may end up with a 1kHz tone instead of the original 9kHz. The output contains something that was never there in the first place – just like the illusionary stationary or backward-turning wheels on the ‘wagon’. Real audio signals, such as voice and music, are generally much more complex than single sinewaves, but the same aliasing processes can occur during sampling. The resulting output may contain tones which were not in the input, resulting in a loss of fidelity. Nyquist rate Fig.10. Waveforms of sampling at exactly the Nyquist rate, example 1. 66 Mathematical analysis of the process of sampling a signal shows that signals with a frequency of less than half the sampling frequency are reliably represented in the sampled data. This is known as the Practical Electronics | May | 2024 Simulation files Most, but not every month, LTSpice is used to support descriptions and analysis in Circuit Surgery. The examples and files are available for download from the PE website: https://bit.ly/pe-downloads rather than 9kHz (that is exactly twice the 1kHz signal frequency). Example results are shown in Fig.10 where it can be seen that the samples exactly align with the peaks in the sinewave. These samples will produce a square wave at 1kHz when passed to a DAC, which will in turn produce a 1kHz sinewave when passed through a suitable reconstruction filter – the sampled signal is successfully recovered. However, Fig.11. Waveforms of sampling at exactly the Nyquist rate, example 2. Fig.11 shows the same situation except Nyquist sampling theorem (also known as the sampling theorem, that the sample pulses are shifted in time (the Tdelay parameter the Nyquist-Shannon theorem and the Whittaker-Nyquistfor the pulse source V2 is changed to 0 instead of 250µs). Shannon sampling theorem, the names honouring people who The sampling aligns with the zero crossing of the sinewave, first published significant work on the topic). The theory shows so all the samples are zero and therefore the sampled signal that if the input does not contain any frequency components cannot be recovered. at or beyond half the sampling frequency then it is possible to perfectly reproduce the original signal from the sample data. Antialiasing filters A sampling frequency of twice the maximum signal frequency If we remove all frequencies at and above half the sampling is called the Nyquist frequency or Nyquist rate. Sampling at frequency from the input before sampling it, then we will less than this frequency is called ‘undersampling’ (which does not have any problems with aliasing. This is the job of the have its uses). Sampling at a rate significantly higher than antialiasing filter. By sampling at just over the Nyquist rate the Nyquist rate is called oversampling and facilitates low it is theoretically possible to perfectly recover the sampled -noise, low-distortion signal processing. Sampling at exactly signal, but doing so implies a close-to-ideal antialiasing filter the Nyquist rate may enable the signal to be recovered but the – we will discuss this in more detail in another article. In results may be wrong for specific alignment of the samples and cases where there is certainty that the sampled signal does not waveform. This is show in the simulation in Fig.9 to Fig.11. The contain frequencies at and above half the sampling frequency schematic (Fig.9) is similar to Fig.5 but with 2kHz sampling an antialiasing filter is not required. Antialiasing filters are not only required in audio and video signal processing; the image sensors in digital cameras are spatially sampling the image projected by the lens onto the sensor. If the image contains small repetitive details aliasing can occur. To overcome this, an optical filter is used to provide an amount of blurring which will prevent aliasing. However, in recent years more digital cameras have become available without anti-aliasing filters. These provide better image quality in situations where aliasing would not occur. NEW! 5-year collection 2017-2021 All 60 issues from Jan 2017 to Dec 2021 for just £44.95 PDF files ready for immediate download See page 6 for further details and other great back-issue offers. Fig.12. The moiré effect in photography is an example of aliasing. It creates artifical stripey patterns when images are captured with repeating patterns that match the spatial frequency of the camera’s sensor pixels. Purchase and download at: www.electronpublishing.com Practical Electronics | May | 2024 67